Case 10 — Cantilever beam with tension-only lateral ties

Description

A vertical cantilever beam of length 1.0 m is fixed at its upper end. Near the base, two horizontal tension-only elements of 1.0 m length each connect the bottom node of the beam to fixed anchors on both sides. A horizontal concentrated load of 1 kN is applied at mid-height of the beam (0.5 m below the fixed support).

Determine:

• Axial force in each tension-only element
• Horizontal displacement at the base node of the beam

Structural scheme

Geometry, boundary conditions, and load applications used in the verification model.

Model parameters

Analytical solution

The system is statically indeterminate. The force in the tie is found from the compatibility condition of deformations:

$$\delta_{D,P} - T \cdot \delta_{11} = T \cdot \frac{L_h}{EA_h}$$

$$T = \frac{\delta_{D,P}}{\delta_{11} + \dfrac{L_h}{EA_h}}$$

where $\delta_{D,P} = \dfrac{5PL^3}{48EI}$ — deflection of point D due to the external load P, calculated on the released cantilever (without the tie);

$\delta_{11} = \dfrac{L^3}{3EI}$ — deflection of point D due to a unit force applied at D;

$L_h,\, A_h$ — length and cross-sectional area of the tie; $E$ — modulus of elasticity; $L$ — height of the cantilever segment BD.

We assume that there is no force in the opposite tie.

$$T = 0.3062\,\text{kN}$$ $$\Delta x(D) = 0.0186\,\text{mm}$$

Numerical results

Displacements

Axial force

Table results from RodX

Comparison

The numerical results obtained with RodX are in agreement with the analytical solution and reference FEA results.